Anti Gaussian Quadrature For Real Definite Integral

Complex Gaussian quadrature for oscillatory integral transforms

The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect to an oscillatory weight, yielding methods with complex quadrature nodes and positive weights. These rules are well suited for highly oscillatory integrals because they attain optimal asymptotic order. We show that for the Fourier oscilla...

Anti-Gaussian quadrature formulas

An anti-Gaussian quadrature formula is an (n+ 1)-point formula of degree 2n− 1 which integrates polynomials of degree up to 2n+ 1 with an error equal in magnitude but of opposite sign to that of the n-point Gaussian formula. Its intended application is to estimate the error incurred in Gaussian integration by halving the difference between the results obtained from the two formulas. We show tha...

Gaussian Quadrature for Kernel Features

Kernel methods have recently attracted resurgent interest, showing performance competitive with deep neural networks in tasks such as speech recognition. The random Fourier features map is a technique commonly used to scale up kernel machines, but employing the randomized feature map means that O(ε-2) samples are required to achieve an approximation error of at most ε. We investigate some alter...

The Gaussian Quadrature Method

Asymptotic estimation of Gaussian quadrature error for a nonsingular integral in potential theory

This paper considers the n-point Gauss-Jacobi approximation of nonsingular integrals of the form ∫ 1 −1 μ(t)φ(t) log(z− t) dt, with Jacobi weight μ and polynomial φ, and derives an estimate for the quadrature error that is asymptotic as n → ∞. The approach follows that previously described by Donaldson and Elliott. A numerical example illustrating the accuracy of the asymptotic estimate is pres...